However flabbily drafted the clause may be, defeat for the government would send a strong signal that Parliament doesn't back the negotiating goals May is pursuing.
Ms Smith said: "I will not vote for any amendment to Brexit legislation that does not deliver the exit from the European Union that my constituents voted for".
"Every time we do anything in the UK, Brussels follows incredibly closely", the minister told the BBC. After a bumpy week of Brexit spats within her administration and with the EU, May wants to fend off another setback in a long-awaited showdown with restive lawmakers.
Potential rebels fell into line after Mr Buckland said ministers were ready to "engage positively" with their concerns before the Bill returns to the Upper House next Monday.
Theresa May ultimately persuaded all but two of her MPs to back her in the decisive vote in Westminster on Tuesday - but she increasingly appears little more than a hostage to the warring factions in a bitterly divided Conservative party.
Parliament must decide whether to support an amendment approved by the House of Lords that could mean sending May back into negotiations with the European Union if lawmakers reject a Brexit deal.
May met with more than a dozen Tory would-be rebels shortly before the vote to reassure them, although exactly what she promised is in dispute.
He confirmed that ministers will seek to overturn 14 amendments which he said would undermine the goal of the Bill and fail to respect the result of the 2016 referendum.
Politics is often about the big picture, but sometimes it is a festival for pedants. Reneging on the promise could have disastrous consequences for the government.
May's Brexit Bill Returns: A Guide to the Main Voting Battles
"If it were to turn out there was a problem, we will deal with it".
May reportedly gave personal assurances to Conservative party rebels, including the former attorney general Dominic Grieve, who then went on to vote with the government and reject the lords amendment on the meaningful vote.
"What it does is put in place a structure if things do go as planned", she said.
"I trust the prime minister. It was the prime minister who I sat in front of this afternoon and who gave us those assurances". Part A said that if parliament rejects the final Brexit deal, the government would have to set out a new approach within seven days. The government fears a weakened negotiating position.
"But I have to say to him, I don't know if members of the House are aware of the headline acts at Labour Live?"
"There is no inconsistency here". We just want to make sure the economy is protected.
An agreement that defused a potential rebellion over handing parliament more control over Britain's exit from the European Union looked in danger of unraveling on Wednesday, when the two camps argued over the shape of a possible compromise on a "meaningful vote".
Earlier on Tuesday the government suffered a setback as one minister resigned over what he called the government's plans to "limit" the role of parliament in shaping Brexit.
Talks with Brussels have stalled over the fraught issue of the Irish border, but both sides are hoping to agree a deal by October in time for the Brexit date of March 29, 2019.